First-row transition metal doped germanium clusters Ge16M: some remarkable superhalogens

A theoretical study of geometric and electronic structures, stability and magnetic properties of both neutral and anionic Ge16M0/− clusters with M being a first-row 3d transition metal atom, is performed using quantum chemical approaches. Both the isoelectronic Ge16Sc− anion and neutral Ge16Ti that have a perfect Frank–Kasper tetrahedral Td shape and an electron shell filled with 68 valence electrons, emerge as magic clusters with an enhanced thermodynamic stability. The latter can be rationalized by the simple Jellium model. Geometric distortions from the Frank–Kasper tetrahedron of Ge16M having more or less than 68 valence electrons can be understood by a Jahn–Teller effect. Remarkably, DFT calculations reveal that both neutral Ge16Sc and Ge16Cu can be considered as superhalogens as their electron affinities (≥3.6 eV) exceed the value of the halogen atoms and even that of icosahedral Al13. A detailed view of the magnetic behavior of Ge16M0/− clusters shows that the magnetic moments of the atomic metals remain large even when they are quenched upon doping. When M goes from Sc to Zn, the total spin magnetic moment of Ge16M0/− increases steadily and reaches the maximum value of 3 μB with M = Mn before decreasing towards the end of the first-row 3d block metals. Furthermore, the IR spectra of some tetrahedral Ge16M are also predicted.


Introduction
Along with silicon, germanium is one of the most important microelectronic materials. The last several decades have witnessed a continuing interest in the clusters of this semiconductor element since their bulk materials can no longer satisfy the current needs of the miniaturization of electronic devices. [1][2][3][4][5][6][7][8][9][10][11] Both silicon and germanium do not favor sp 2hybridization such as carbon whose small clusters tend to form linear or planar cyclic structures, but rather prefer 3D species arising from a tetrahedral sp 3 hybridization. Consequently, pure germanium clusters with high symmetry are oen unstable in the form of empty caged structures. Since the rst observation of the far more abundant Si 15 M and Si 16 M that was reported in 1987 from the laser photoionization time of ight mass spectra, 12 subsequent investigations using ab initio calculations on the geometrical and electronic structures of the Si 15 and Si 16 clusters doped with several transition metals such as Cr, Mo, and W have been performed. 13 These studies showed that the transition metal dopant is oen located inside a polyhedral cage forming the Si 15 M and Si 16 M clusters that have high thermodynamic stability and low magnetic moments as compared the M metal dopants. In a recent review article, Kumar et al. 14 analyzed in detail the electronic and geometrical structures of the Si 15 and Si 16 clusters doped with several transition metals and showed that the transition metal atom is always endohedrally doped within the silicon cage. This thus demonstrates that introduction of hetero-atoms as dopants into hollow cages can supply us with a valuable pathway to stabilize endohedral cage-like clusters as well as to adjust their many novel physico-chemical properties. Motivated by such a fundamental feature, along with several studies performed on doped silicon clusters, a large number of both experimental and theoretical investigations on germanium clusters doped by various chemical elements have been carried out.  Transition metal atoms that have unpaired valence electrons in their nd electronic congurations, are inherently magnetic elements. They have been considered as interesting dopants in clusters since interactions between these impurities and the host are expected to alter both electronic and geometrical structures and thereby to generate the doped clusters possessing some novel physico-chemical properties. Moreover, as stated above, due to their high coordination number, transition metals can endohedrally be doped and stabilize the caged structures and simultaneously tailor magnetic properties of host clusters. Indeed, previous studies of singly transition metal doped germanium clusters showed that starting from the size n ¼ 9, the Ge unit absorbs the Ni and Ru dopant endohedrally in giving rise to the most stable isomers of Ge n Ni and Ge n Ru. 22,24 The metal atom is encapsulated inside a germanium cage at n ¼ 10 when the dopant is Ti, V, and Cu, 20,25,29 and the critical size for the heavy metal W atom being completely enclosed into a caged germanium framework in the Ge n W clusters turns out to be at n ¼ 12. 33 A theoretical investigation 34 on divalent-metal atom doped silicon, germanium and tin clusters X n M (X ¼ Si, Ge, Sn; n ¼ 8-12 and 14) demonstrated that the 12-and 14-atom clusters can be transformed into magic clusters upon doping. Particularly, the manganese-doped X 12 Mn was found to be an icosahedral superatom with a high magnetic moment of 5 m B . 34 In an examination of doubly iron-doped germanium clusters, Liang and co-workers also indicated that both neutral and cationic states of Ge n Fe 2 0/À adopt polyhedral cage-like shapes with one Fe atom located inside the cage with 9 # n # 12. 26 Soon aer the theoretical prediction of metal-encapsulated silicon cages, 40 Kumar and Kawazoe performed a series of calculations to explore the possible germanium cages stabilized by metal doping. Analogous to M@Si n clusters, they explored M@Ge n (n ¼ 14-16 and M ¼ Ti, Zr, Hf, Fe, Ru, Os) clusters with various possible cage congurations such as the Frank-Kasper (FK) polyhedron, capped decahedron, fullerene-like cage and cubic cage. [41][42][43][44] Remarkably, the Ge 16 16 Zr is even larger than the value for the lowest-energy isomer of FK Si 16 Zr discovered before. 40 Most recently, Du and coworkers carried out an investigation on the interaction in dimers of well-known endohedrally doped clusters, including several X 16 M clusters with X being the tetravalent elements, and found that Ge 16 Ti cage clusters emerge as suitable building blocks to assemble generating solids and nanostructures with enhanced stabilities and diverse physical properties. 45 To date, many transition metal-silicon and transition metalgermanium clusters have been examined, and the understanding on the cage-like silicon and germanium structures stabilized by doping of some transition metals has well been established on the basis of the concept of lling the electron shells for superatoms within a spherical potential model, and also of various electron counting rules including the Wade-Mingos rules, systems with 18 and 32 electrons. 14 However, to the best of our knowledge, until recently only one study on the trimeric Ge 2 M including all 3d transition metals M was reported. 46 Therefore, systematic theoretical studies on a certain series of transition metal-germanium clusters are still necessary in order to understand more deeply the relationship between structures and electronic properties of the transition metal doped Ge cluster, especially the Ge 16 ones bearing the characteristic Frank-Kasper geometry. In this context, we set out to perform a systematic theoretical investigation on the geometries, stability, and magnetic properties of the germanium clusters doped by one atom belonging to the 3d row transition metals in both neutral and anionic states Ge 16 M 0/À , with M going from Sc to Zn. Using density functional theory (DFT) calculations, we thoroughly determine the geometries of the lowest-lying equilibrium structures and thereby explore their structural evolution, as well as assign their electronic congurations, energetic parameters and magnetic properties. In particular, some systems behaving as strong superhalogens are discovered.

Computational methods
On the basis of a reliability test that has been obtained from a previous study on germanium-based clusters, 21 we select the hybrid B3PW91 functional in conjunction with the 6-311+G(d) basis sets as implemented in Gaussian 09 package 47 for all electronic structure calculations carried out in this work. The unrestricted formalism is used for species with an open electronic shell. The search for local energy minima is conducted using the two approaches. First, plausible structures of Ge 16 M clusters are generated using a stochastic algorithm, 48 which was improved based on the random kick procedure reported by Saunders. 49 By another way, initial structures of each Ge 16 M are manually constructed by adding the M atom at all possible positions on the surfaces of the reported low-lying isomers of Ge 16 . 10,11 The initial guess structures are then geometrically optimized using the hybrid B3PW91 functional in conjugation with the small LANL2DZ basis set. Several local minima obtained by different approaches turn out to be identical. The local energy minima having relative energies of <5 eV with respect to the lowest-lying isomer are then reoptimized at different spin states using the same B3PW91 functional but with the larger 6-311+G(d) basis set. Harmonic vibrational frequencies and zero-point energy (ZPE) corrections of the Ge 16 M clusters are subsequently calculated at the same level. Unless otherwise stated, relative energies quoted hereaer are determined from B3PW91/6-311+G(d) + ZPE computations.
Furthermore, the natural bond orbital (NBO) analyses are performed by using the NBO 3.0 program implemented in the Gaussian package to examine the electronic conguration and thereby rationalize the magnetic and chemical bonding properties of the clusters considered. Based on the NBO analyses, the magnetic moments including the total (TMMs) and local (LMMs) values are dened as the difference between the numbers of spin-up and spin-down electrons occupying the molecular/atomic orbitals of the cluster/atom.

Geometrical structures
Shapes of the optimized equilibrium structures of both neutral and anionic series of Ge 16 M clusters, their spin states, and DFT relative energies are shown in Fig. 1 and 2. Because of the large number of isomers located on the potential energy surface of each cluster, only some low lying isomers whose relative energies are close to the corresponding ground state structure are presented for each dopant M.
As for a convention, a X.M.Y label is used to denote the isomers considered, in which X ¼ n and a stand for a neutral and anionic state, respectively, M ¼ Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn, and Y ¼ A, B, C. refers to the different isomers with increasing relative energy. Thus, X.M.A invariably refers to the lowest-energy isomer of the X.M series.
The main characteristics of the geometrical features can briey be summarized as follows: For M ¼ Sc, Ti and V, our calculated results are in good agreement with the previous studies. 36,41 The most stable Ge 16 M in both neutral and anionic states consistently prefer a FK structure in which the dopant atom is endohedrally located at the central position of the FK Ge 16 cage. Remarkably, the lowestlying isomers of both isoelectronic Ge 16 Sc À anion and Ge 16 Ti neutral are much more stable than the next isomers with large relative energy gaps of 0.84 and 0.95 eV, respectively. The most stable Ge 16 V in both neutral and anionic states still retain the FK form, but their relative energy gaps decrease to <0.5 eV. Besides, the low-lying isomers of Ge 16 V À anion are found to exist in the triplet state. For Ge 16 Cr, the FK is no longer the most stable form, as they are 0.24 and 0.14 eV higher in energy than the neutral and anion, respectively, of another endohedral structure in which the Cr atom is encapsulated in a C 3v cage. Ge 16 Cr also favors high spin multiplicity, corresponding to the triplet and quartet states for the neutral and anion, respectively.
The three lowest lying isomers of the neutral Ge 16 Mn prefer endohedrally doped structures and are stable in a quartet state. n.Mn.A becomes 0.25 eV more stable than the other cage n.Mn.B. In the anionic state, however, a competition in energy among three most stable isomers emerges with relative energy gaps of <0.1 eV. Particularly, the a.Mn.A, constructed from fusion of two Ge 10 in which the endohedral Ge atom, or the Ge atom at the vertex of the lower Ge 10 block, is substituted by the Mn atom, 48 and the FK a.Mn.B, are energetically degenerate within a small energy difference of only 0.02 eV, and both of them have a magnetic moment of 4 m B , arising from a quintet spin state. Remarkably, the geometry of a.Mn.A is also retained as the most stable one for all remaining Ge 16 M À anions, with M being Fe, Co, Ni, Cu and Zn, despite a competition in energy in Ge 16 Co À where the triplet state a.Co.A is only 0.01 eV lower in energy than the singlet a.Co.B. Accordingly, both Co derivatives a.Co.A and a.Co.B are energetically degenerate.
The shapes of the low lying isomers of the neutral Ge 16 Fe, Ge 16 Co, and Ge 16 Ni are similar to that of the Ge 16 Mn. However, optimization calculations indicate that the two endohedral isomers n.Ni.A and n.Ni.B of Ge 16 Ni are again practically degenerate with a negligible energy gap of 0.01 eV. Finally, the lowest-lying isomers of Ge 16 Cu and Ge 16 Zn exhibit the same shape in both neutral and anionic states.

Stabilities
In order to probe the inherent thermodynamic stability of the Ge 16 M clusters considered, their average binding energies (E b ) are examined and compared to those of the relevant pure germanium clusters Ge 17 in both neutral and anionic states. The E b values of the Ge 16 M clusters can conventionally be dened in eqn (1) and (2): (2) where E(Ge), E(Ge À ), and E(M), are the total energies of the Geatom, the anion Ge À , and the M-atom, respectively. E(Ge 16 M) and E(Ge 16 M À ) are the total energies of the neutral and anionic of Ge 16 M, respectively. Similarly, for the neutral Ge 17 and anionic Ge 17 À , the E b can be dened by eqn (3) and (4), respectively, as follows: where E(Ge 17 ) and where E(Ge 17 À ) are the total energies of the pure neutral and anionic Ge 17 , respectively, that were reported in the previous studies. 10,11 All these energy values are obtained from B3PW91/6-311+G(d) + ZPE calculations and the plots of E b depicted in Fig. 3a 16 Zn takes the lowest E b values in both neutral and anionic states. These calculated results prove that a doping of the rst-row transition metal M, except for Cr, Mn, Cu and Zn, into Ge 16 enhances the cluster stability as compared to the pure germanium clusters Ge 17 in both neutral and anionic states. Remarkably, the neutral Ge 16 Ti and anionic Ge 16 Sc À , which possesses each 68 valence electrons, reveal the highest E b values as compared to the remaining Ge 16 M counterparts.
To reinforce the above ndings, we further examine the embedding energy (EE) of the clusters considered. Embedding energy is dened as the energy gained in incorporating a Mdopant into the Ge 16 hosts and dened by eqn (5): where E(Ge 16 0/À ) are the total DFT energies of the neutral and anionic Ge 16 clusters, respectively. These total energies are calculated for the ground states of the pure clusters Ge 16 0/À which were previously reported. 10,11 Fig. 3b indicates that both Ge 16 Ti and Ge 16 Sc À are really characterized by the highest EE values for the neutral Ge 16 M and anionic Ge 16 M À clusters, respectively. These predictions are in good agreement with the E b values mentioned above, and it can thus be concluded from these observations that an enhanced thermodynamic stability is established for both isoeletronic Ge 16 Ti and Ge 16 Sc À species.
The E b values of all anionic Ge 16 M À and pure Ge 17 À clusters are obviously higher than those of the neutral counterparts, as shown in Fig. 3a 16 Sc can be interpreted in the same way as that applied to the neutral Al 13 , which is well-known for its very large electron affinity exceeding that of halogen atoms and has thus been named as a superhalogen. 50 Similarly, the neutral Ge 16 Sc, as formed from the detachment of one electron from the closed-shell structure of the anion Ge 16 Sc À possessing an enhanced thermochemical stability, also has a very large electron affinity. As stated above, calculations reveal that the EA of Ge 16 Sc amounts to 3.8 eV, which is even larger than that of 3.6 eV of Al 13 . 51 In contrast to Ge 16 Sc, the following neutral member Ge 16 Ti, which is stabilized by a closed shell lled by 68 valence electrons in a singlet state, exhibits the smallest EA value due to the low stability of the corresponding anion. In this context, Ge 16 Sc can be considered as a superhalogen.
The enhanced stability of both isoelectronic Ge 16 Ti and Ge 16 Sc À FK structures can be rationalized by examining their MO pictures under the viewpoint of the electronic shells in a Jellium model (JM), 52 which is successfully applied to clarify the stability of various structural motifs of atomic clusters in previous studies, particularly those based on Group IVa atoms. 48,53,54 According to this simple model, the valence electrons are freely movable in a simple mean-eld potential constructed by the nuclei of atoms; the valence electrons ll the orbitals following the pattern of atomic orbitals (AO) as [1S 2 1P 6 1D 10 2S 2 1F 14 2P 6 1G 18 2D 10 .] corresponding to the numbers of electrons of 2, 8, 18, 20, 34, 40, 58 and 68, etc., that emerge as the magic numbers consistent with a complete lling of the successive electronic shells. As a consequence, a cluster that possesses a valence electron number belonging to this magic number series is able to attain an enhanced thermodynamic stability.
This corresponds to a sequence of electronic shell model as: [1S 2 1P 6 1D 10 1F 14 2S 2 1G 2 2P 6 1G 16 2D 10 ] For both clusters, the lowest-lying MOs include an s-type valence orbital of the 1S and three p-type orbitals of the 1P subshells. The MOs of both 1D and 1F subshells are mainly composed of s-AO of Ge and the remaining AOs of both germanium and the transition metal dopant with much smaller contributions. For the neutral Ge 16 Ti, the MO of 2S subshell are principally formed by the s-AO(Ti), whereas the MO of 2S of anion Ge 16 Sc À are constructed by interaction between s-AO(Sc) and p-AOs(Ge). The MOs of 2P and 1G subshells are constructed by a combination of both s-and p-AOs of Ge-atoms. Finally, the 2D subshell is composed of both s-and p-AO of Ge and d-AO of the metal impurity. In general, the electronic conguration of the both neutral Ge 16 Ti and anion Ge 16 Sc À basically satises the electronic shell model of [1S 2 1P 6 1D 10 1F 14 2S 2 1G 2 2P 6 1G 16 2D 10 ] and makes them the enhanced stability species with a magic number of 68 valence electrons.
Remarkably, in addition to the nding of Ge 16 Sc superhalogen, calculations reveal that the ground state of the anion Ge 16 Cu À also has a closed electronic shell and high stability as compared to its neutral form, and being much more stable than the second isomer with a large relative energy gap of 0.69 eV. This is caused by the fact that the neutral Ge 16 Cu has a large electron affinity of 3.6 eV, again approximate to those of the chlorine atom and Al 13 mentioned above. Accordingly, Ge 16 Cu can also be considered as a superhalogen. The large EA of Ge 16 Cu can be interpreted based on MO approaches. Of the Ge 16 M clusters in both neutral and anionic forms, the electronic structure of the anion Ge 16 Cu À can be considered as a closedshell by 68 electrons in the pool of valence electrons of the whole cluster such as in the case of Ge 16 Sc À and Ge 16 Ti even though it possesses a non-spherical like geometry. This could be rationalized by considering the following fact. Each Ge delocalizes four electrons whereas the Cu dopant delocalizes three of its eleven 4s 1 3d 10 valence electrons including one 4s and two 3d into the pool, and while the added electrons are also delocalized in the shell of the entire cluster, the eight remaining 3d electrons are localized in four 3d orbitals of the central Cu.
The reason for such a behaviour of the Cu atom in the Ge 16 Cu À anion is that it is located at the trigonal-prismatic hole formed by six nearest Ge atoms. Following the ligand eld effects induced by the trigonal-prismatic coordination within a C 3v point group of the whole Ge 16 Cu À cluster, the ve degenerate 3d AOs of the Cu atoms split into three groups of Eðd xz ; d yz Þ; Eðd x 2 Ày 2 ; d xy Þ and Aðd z 2 Þ irreducible representations. While the two former groups have lower energies, the latter possessing higher energy matches the energies of the valence AOs of the Ge neighboring atoms, and thereby combine well with them to form shell MOs of the resulting cluster. Based on shapes and relative energies of the MOs, we can now assign the electronic energy levels of the anion Ge 16

Electron shell of 68 valence electrons and Jahn-Teller distortion
Let us now describe in some detail the electronic congurations of the neutral Ge 16 Ti as well as the anionic Ge 16 Sc À that possess FK shape in T d symmetry. Each Ge atom in Ge 16 Ti contributes four valence electrons, whereas the Sc and Ti atom contribute three and four, respectively, to its cluster shell. The number of valence electrons contributed by the constitution atoms to the cluster shell amounts to 68 that occupy thus 34 shell MOs. As described in Section 3.2 above, the electronic conguration of this cluster can be written as [1S 2 1P 6 1D 10 2S 2 1F 14 2P 6 1G 18 2D 10 ] in which the HOMO is the 2D shell and the LUMO is the 1H. In a T d point group, the D shell orbitals of either Ge 16 Ti or Ge 16 Sc À split into 2-fold and 3-fold degenerate orbitals, corresponding to an E irreducible representation and a T irreducible representation, respectively, or it could be written as D ¼ E + T. The F shell orbital of both Ge 16 Ti and Ge 16 Sc À split to 1-fold and 3-fold and 3-fold degenerate orbitals, namely F ¼ T + T + A. In the same vein, the G shell orbital splits to 1-fold and 3-fold, 3fold and 2-fold degeneracy orbitals, namely G ¼ A + 2T + E. Therefore, the electron shell conguration of each cluster could be written as follows: [1S 2 1P 6 1D 2-folds 4 1D 3-folds 6 2S 2 1F 3-folds 6 1F 3-folds 6 1F 1-fold 2 2P 6 1G 1-fold 2 1G 3-folds 6 1G 3-folds 6 1G 2-folds 4 2D 10 ].
A closer look at the shapes of shell MOs shows that the ordering of some shell MOs alters, and the shell electron conguration of the cluster becomes: [1S 2 1P 6 1D 2-folds 4 1D 3folds 6 1F 1-fold 2 1F 3-folds 6 1F 3-folds 6 2S 2 1G 1-fold 2 2P 6 1G 3-folds 6 1G 3folds 6 1G 2-folds 4 2D 2-folds 4 2D 3-fold 6 ] as this can be seen in Fig. 6a and c. Nevertheless, geometry optimizations reveal that only FK structures with a closed shell lled by 68 valence electrons, including neutral Ge 16 Ti and anionic Ge 16 Sc À clusters, possess a T d high symmetry whereas the remaining FK structures, which enclose more or less than 68 valence electrons, exist at lower symmetry. Such a geometrical distortion can be understood by the Jahn-Teller effect. Let us rst examine the open-shell neutral Ge 16 Sc having 67 valence electrons due to the fact that the HOMO of the anion Ge 16 Sc À is a 2D shell orbital; the latter has the value of azimuthal quantum number l equal to 2 and a 5-fold degeneracy. In the T point group, the 2D shell orbitals are reduced to the E + T irreducible representations in which the T orbitals, or the 2D three-fold levels are higher energetically. As an electron is removed from such a HOMO, namely the 2D three-fold of the anionic Ge 16 Sc À , to form the neutral Ge 16 Sc, the corresponding SOMO is triply degenerate and the T d structure of the Ge 16 Sc neutral is subjected to a distortion to a C 3v group, accompanying a splitting of the T MOs, or the 2D three-fold one, into E + A 2 orbitals, and the A 2 orbital is now singly occupied (cf. Fig. 6b).
In the case of 69 electrons of the Ge 16 Ti À anion, it is worth to note that the LUMO of the Ge 16 Ti neutral cluster corresponds to a 1H shell orbital which has the value of azimuthal quantum number l of 5 and a 11-fold degeneracy. In the T point group, the 1H shell is reduced into E + 3T irreducible representations. In going from the neutral Ge 16 Ti to the anionic Ge 16 Ti À , the incoming electron lls in one of the degenerate LUMO of T representation, which thus causes a distortion of the T d structure of the Ge 16 Ti neutral, again to a C 3V structure of the resulting Ge 16 Ti À anion, accompanying with a splitting of the T MOs into E + A 2 orbitals, of which the A 2 orbital accommodates the unpaired electron (cf. Fig. 6d).
It is also worth mentioning that the vertical electron detachment energy between the neutral Ge 16 Sc and its anion at the geometrical structure of the anion which has 68 electrons, amounts to 3.95 eV, whereas the adiabatic detachment energy is 3.81 eV. Thus, the energy gain of the Ge 16 Sc neutral upon distortion is 0.14 eV. Similarly, the energy gain of Ge 16 Ti À due to a distortion from T d to C 3v symmetry is calculated at a value of 0.23 eV. Such a small but signicant amount of energy originates in an intrinsic instability of the T 1 or the 2D three-fold 1 electron conguration of the HOMOs of Ge 16 Sc, as well as the T 1 or 1H three-fold 1 conguration of the LUMOs of Ge 16 Ti À , all in T d symmetry. The latter are therefore distorted to C 3v point group yielding such an energy gain of the system. The SOMO-LUMO gap of 1.1 eV in the 67 electrons neutral Ge 16 Sc is very close to the SOMO-LUMO gap in the 69 electrons of anionic Ge 16 Ti À . This reects the fact that both systems undergo a comparable symmetry reduction when going from the Ge 16 Sc À to Ge 16 Sc as well as from Ge 16 Ti to Ge 16 Ti À , as this is illustrated in Fig. 6b and d. In summary, two forces are combined in the formation of the C 3v structures of 67-electron Ge 16 Sc neutral and 69 electron Ge 16 Ti À anion: the major intrinsic stability of the 68 electron Ge 16 Sc À and Ge 16 Ti counterparts which favor a T d symmetry, and the reorganization energy gained during the symmetry lowering from T d to C 3v point group. This also makes the average binding energy of Ge 16 Sc À much higher than that of its neutral, while the average binding energy of Ge 16 Ti À is only slightly higher.

Spin magnetic moments
It is typical that when a metallic cluster is doped by a transition metal atom, the outer-most orbitals of the impurity including d and s shell can combine with the valence orbitals of the host to form shell orbitals of the resulting doped clusters. For the Fig. 6 MO diagrams of (a) Ge 16 Sc À anion; (b) Ge 16 Sc 0 neutral; (c) Ge 16 Ti 0 neutral; and (d) Ge 16 Ti À anion. Black arrow represents electron. Bright blue lines represent the 1S and 2S shell orbitals; orange for the 1P and 2P shell orbitals; dark blue for the 1D and 2D shell orbitals; green for the 1F shell orbitals; violet for the 1G shell orbitals, and red lines represent for HOMO of the clusters.
investigated Ge 16 M, as each Ge atom delocalizes its 4 valence electrons, the corresponding Ge 16 M À anion can approach 68 electrons if the metal dopant could delocalize its 3 valence electrons. The remaining valence electrons of the metal will combine to build up its magnetic moment. Fig. 7 shows the change in the total spin magnetic moment (TMM) value of Ge 16 M in both neutral and anionic states as M goes from Sc to Zn. The perception of how the total and local spin magnetic moments of the clusters arise can be conrmed in considering the calculated total and local spin magnetic moments on each constituent atoms of the clusters that are listed in Table 1. The total spin magnetic moment of the Ge 16 M À anion increases steadily from 1 m B for the Ti dopant to 4 m B for the Mn dopant, then decreases to 1 m B for the Ni dopant. For the Ge 16 M À anions, their magnetic moments are mostly held on the transition metal atoms, being 0.4, 1.5, 2.9, 3.1, 2.2, and 1.4 for the Ti, V, Cr, Mn, Fe, and Co dopants, respectively. Exceptions include the Ge 16 Ni À and Ge 16 Zn À in which the total magnetic moments are delocalized all over the entire skeleton.
In order to gain more insight into the spin magnetic behavior of Ge 16 M clusters, along with Table 1 listing total and local spin magnetic moments, the total density of states (TDOS) and partial density of states (PDOS) of the anionic Ge 16 M À clusters from the Ti to Ni dopants are plotted in Fig. 8. Spin-up and spin-down densities of states are plotted separately on the same graph for each of the clusters. It has been stated that the relative shi between the spin-up and spin-down bands indicates the degree of spin-exchange splitting; the larger the shi of DOS bands, the larger the magnetic moment of the cluster. 55 As we glance at Fig. 8, we can realize that the shi is large for the Ge 16 M À (M ¼ Cr, Mn, Fe, Co) clusters suggesting that they possess high spin magnetic moments, while it is slight in the cases of Ge 16 Ti À and Ge 16 Ni À clusters.
The spin magnetic moment of the whole cluster is mainly created by the unpaired electrons, and the spin magnetic moments localized on the metal dopant arise from the density of unpaired electron contained in its orbitals or the difference in partial density of aand b-electrons of the M atom. For the Ti and V clusters ( Fig. 8a and b), the a-HOMO states are located at distinctive energy level as compared to the a-, b-inner states, while for the other Ge 16 M, with M being Cr, Mn, Fe, Co, and Ni, the a-HOMO states are situated in almost the same energy region as with the a, and b-inner states. For the DOS of Ge 16 Cr À anion, in the domain of states ranging from À4.3 to À2.0 eV, there is an obvious presence of the density of a-state of d orbitals of Cr (the blue curve) without the b-state counterpart. This information implies that the total magnetic moment of the cluster is mainly dominated by Cr-d states, while Ge-s and Ge-p states make rather a small contribution. A similar argument can be made for the successive clusters. For Ge 16 Mn À in the domain from ca. À6.5 to À2.0 eV there are densities of a-states of s-and d-orbitals of Mn (the green and blue curves) without their bstate counterparts; for Ge 16 Fe À in the domain from ca. À7.0 to À2.0 eV there are densities of a-states of s-and d-orbitals of Fe (the green and blue curves) without their b-state counterparts; for Ge 16 Co À in the domain from ca. À7.0 to À2.0 eV there are densities of a-states of s-and d-orbitals of Co (the green and blue curves) without their b-state counterparts. This observation is in line with the calculated local magnetic moments on the Cr, Mn, Fe, Co atoms which amount to 2.9, 3.1, 2.2 and 1.4 m B , respectively. Thus, although the Ge 16 cage somehow quenches the usually large magnetic moments of free transition metal atoms, the latter property remain substantial in the doped derivatives.

IR spectra
As discussed above, several lower-energy structural and spin isomers for each Ge 16 M cluster are considered, and we report herein only the lowest-energy isomers for certain species. In some cases, the energy difference of the most stable isomers is really small that cannot allow us to distinguish the ground state structure. Moreover, no infrared (IR) spectrum of any Ge n M has been reported both experimentally and theoretically so far. The IR spectra are expected to provide us with a ngerprint for assignment of the cluster in terms of their metallic dopant, especially for the geometrical structure. For the purpose to help for distinguishing low-lying energy isomers, the calculated vibrational spectra of two lowest-lying isomers for some selected Ge 16 M, namely the Ge 16 Sc and Ge 16 Ti in both neutral and anionic forms, are plotted in Fig. 9. The vibrational frequency range goes from 0 to 350 cm À1 , as no signals are found at higher photon energy. As it could be seen in Fig. 9a and b, the vibrational spectra for the ground state of Ge 16 Sc À anion and Ge 16 Ti neutral, each has 68 valence electrons and is in T d symmetry, are relatively simple featuring a highly intense peak, centered at $304 and 292 cm À1 with T degenerate modes, respectively.
Unlike the IR spectra of the ground states, those of the next isomers turn out to be more complicated and characterized by several highly intense peaks in the range of 260 to 300 cm À1 (see Fig. 9c and d). The following frequencies can be noted. While for the ground state of Ge 16 Sc neutral, isomer n.Sc.A has n stretching Sc-Ge being of $298 cm À1 , the n stretching Sc-Ge of the next isomer n.Sc.B is of $280 and 290 cm À1 . For the Ge 16 Sc À anion, FK structure a.Sc.A has a n stretching Sc-Ge of $304 cm À1 , while n stretching Sc-Ge of the isomer a.Sc.B is of $264, 279 and 289 cm À1 . Although for Ge 16 Ti, FK structure n.Ti.A only has one peak at 292 cm À1 , being three-fold degenerate Sc-Ge stretching mode, for the next isomer n.Ti.B, the Sc-Ge stretching modes appear at three frequencies $263, 275 and 298 cm À1 . For the anion Ge 16 Ti À , a.Ti.A in distorted FK shape (C 3v ) leads its n stretching Sc-Ge being of $246 and 281 cm À1 , which is a two-fold degenerate mode E, while a.Ti.B isomer has a n stretching Sc-Ge being of $252, 270 and 279 cm À1 . The large difference in vibrational spectra for both nearly degenerate isomers and their high intense peaks can be used to assign the ground state structure when they can be generated experimentally and characterized spectroscopically.
By analysis of their IR spectra, a symmetry lowering of the cluster in going down from 68 to 67 as well as going up to 69 valence electrons can be recognized. The IR spectrum of Ge 16 Ti is characterized by a single peak centered at 292 cm À1 , corresponding to the vibrational modes of Ti atom inside the Ge 16 cage. Although the cluster has 45 vibrational modes in total, only the vibrational modes of the Ti atom inside the cage are IR active with notably high intensity. Other modes that correspond to deformation of the Ge 16 cage do not result in signicant IR intensity. The stretching modes of the Ti atom inside the Ge 16 cage belong to the T 2 irreducible representation. As the cluster receives one electron to form the Ge 16 Ti À , the anion is distorted to C 3v point group and the n(Ti-Ge 16 ) T 2 mode is reduced to the E + A 2 modes. As this could be seen in Fig. 9, the difference of Table 1 Total spin magnetic moment (TMM, m B ) of Ge 16 M and local spin magnetic moment (LMM, m B ) of each atom at (a) neutral and (b) anionic states. Atom number and corresponding Cartesian coordinates of each atom in Ge 16 M are given in the  22 cm À1 between the E and A 2 modes is relatively signicant in view of the low frequency. It is worth noting that for Ge 16 Ti À , its A 2 n(Ti-Ge 16 ) stretching mode has a lower frequency than the E n(Ti-Ge 16 ) stretching. The added electron in the Ge 16 Ti À anion causes the cluster to distort from T d to C 3v point group, giving rise to a lowering of electron density along one of the three n(Ti-Ge 16 ) stretching modes.

Concluding remarks
In the present theoretical study, the geometric and electronic structures, thermodynamic stability, and magnetic properties of the 16-atom germanium clusters doped with the rst-row 3d transition metal atoms, Ge 16 M with M ¼ Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn, in both neutral and anionic states, were investigated using quantum chemical (DFT) methods. The most stable isomers of Ge 16 M, as M goes from Sc to V, prefer a Frank-Kasper (FK) structure in which the metal dopant is endohedrally encapsulated at the central position of a Ge 16 FK cage. In particular, both the anionic Ge 16 Sc À and neutral Ge 16 Ti whose electronic shells are lled by a magic number of 68 valence electrons, are characterized by a perfect FK tetrahedral geometry and enjoy an enhanced thermochemical stability with high average binding energies and embedding energies. Their higher stability can be interpreted in terms of the electronic shells of the Jellium model. Analyses of electronic conguration also indicate that the geometric distortions from an FK tetrahedron Ge 16 M having more or less than 68 valence electrons are caused by a Jahn-Teller effect arising from the degenerate frontier orbitals.
Perhaps most interestingly is the result obtained from energy calculations that revealed that both neutrals Ge 16 Sc and Ge 16 Cu emerge as superhalogens, due to a characteristic that each possesses a large electron affinity of 3.8 and 3.6 eV, respectively. These electron affinities exceed the values of halogen atoms and even that of the well-known superhalogen Al 13 ($3.6 eV). Moreover, a comprehensive picture of the magnetic behavior is displayed for Ge 16 M clusters at both neutral and anionic states, in which the observed dopant-dependent magnetic moment can be understood by a charge distribution analysis. As M goes from the le to the right side on the rst-row transition metal atoms in the Periodic Table, corresponding to Sc to Zn, the total magnetic moment of Ge 16 M rst takes a low value at M ¼ Sc and Ti, then increases steadily and reaches the maximum value of 3 m B at M ¼ Mn, before decreasing towards the end of the row due to the fact that these magnetic moments are mostly held on the metal dopants. This result opens up an avenue that a magnetically inert germanium cluster can be induced to a relatively high magnetic moment following doping by a suitable transition metal impurity. Finally, the IR spectra of FK Ge 16 M are simulated as a helpful guide for future experimental assignment of these degenerate ground state clusters.

Funding information
This work is funded by VinGroup (Vietnam) and supported by VinGroup Innovation Foundation (VinIF) under project code VinIF.2020.DA21. Fig. 9 The IR spectra of: (a) Ge 16 Ti 0/À and Ge 16 V. The red curve represents for IR spectrum of Ge 16 V neutral, the green curve for Ge 16 Ti À anion, and the dark blue for the Ge 16 Ti neutral; (b) Ge 16 Sc 0/À clusters. The red curve represents for the Ge 16 Sc neutral, and the blue for the Ge 16 Sc À anion; (c) the two lowest-energy isomers of the Ge 16 Sc 0/À ; (d) the two lowest-energy isomers of the Ge 16 Ti 0/À .